Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

Topic: The Distinguishability argument of the Reals.
Replies: 83   Last Post: Jan 7, 2013 12:58 AM

 Messages: [ Previous | Next ]
 Virgil Posts: 8,833 Registered: 1/6/11
Re: The Distinguishability argument of the Reals.
Posted: Jan 4, 2013 4:31 PM

In article <kc6idc\$jc\$1@Kil-nws-1.UCIS.Dal.Ca>,
gus gassmann <gus@nospam.com> wrote:

> On 03/01/2013 5:53 PM, Virgil wrote:
> > In article
> > WM <mueckenh@rz.fh-augsburg.de> wrote:
> >

> >> On 3 Jan., 14:52, gus gassmann <g...@nospam.com> wrote:
> >>

> >>> Exactly. This is precisely what I wrote. IF you have TWO *DIFFERENT*
> >>> reals r1 and r2, then you can establish this fact in finite time.
> >>> However, if you are given two different descriptions of the *SAME* real,
> >>> you will have problems. How do you find out that NOT exist n... in
> >>> finite time?

> >>
> >> Does that in any respect increase the number of real numbers? And if
> >> not, why do you mention it here?

> >
> > It shows that WM considerably oversimplifies the issue of
> > distinguishing between different reals, or even different names for the
> > same reals.

> >>>
> >>> Moreover, being able to distinguish two reals at a time has nothing at
> >>> all to do with the question of how many there are, or how to distinguish
> >>> more than two. Your (2) uses a _different_ concept of distinguishability.-

> >>
> >> Being able to distinguish a real from all other reals is crucial for
> >> Cantor's argument. "Suppose you have a list of all real numbers ..."
> >> How could you falsify this statement if not by creating a real number
> >> that differs observably and provably from all entries of this list?

> >
> > Actually, all that is needed in the diagonal argument is the ability
> > distinguish one real from another real, one pair of reals at a time.

>
> Exactly. The only reals that matter to Cantor's argument are the
> *countably* many that are assumed to have been written down. There is no
> need (nor indeed an effective way) to distinguish the constructed
> diagonal from *all* the potential numbers that could have been
> constructed that are not on the list, either. Any *one* number not on
> the list shows that the list is incomplete and thus establishes the
> uncountability of the reals.

But try getting WM to see it!
--

Date Subject Author
1/1/13 Zaljohar@gmail.com
1/2/13 mueckenh@rz.fh-augsburg.de
1/2/13 Virgil
1/3/13 Virgil
1/3/13 Zaljohar@gmail.com
1/3/13 gus gassmann
1/3/13 Zaljohar@gmail.com
1/3/13 gus gassmann
1/3/13 Zaljohar@gmail.com
1/3/13 mueckenh@rz.fh-augsburg.de
1/3/13 Virgil
1/3/13 fom
1/4/13 Zaljohar@gmail.com
1/4/13 fom
1/3/13 mueckenh@rz.fh-augsburg.de
1/3/13 Virgil
1/3/13 fom
1/3/13 Virgil
1/4/13 gus gassmann
1/4/13 mueckenh@rz.fh-augsburg.de
1/4/13 fom
1/5/13 mueckenh@rz.fh-augsburg.de
1/5/13 Virgil
1/5/13 fom
1/4/13 Virgil
1/5/13 mueckenh@rz.fh-augsburg.de
1/5/13 Virgil
1/4/13 Virgil
1/4/13 gus gassmann
1/4/13 ross.finlayson@gmail.com
1/5/13 Virgil
1/5/13 ross.finlayson@gmail.com
1/5/13 Virgil
1/5/13 fom
1/5/13 ross.finlayson@gmail.com
1/6/13 fom
1/6/13 ross.finlayson@gmail.com
1/6/13 Virgil
1/6/13 ross.finlayson@gmail.com
1/6/13 Virgil
1/6/13 ross.finlayson@gmail.com
1/6/13 Virgil
1/6/13 ross.finlayson@gmail.com
1/6/13 Virgil
1/6/13 ross.finlayson@gmail.com
1/6/13 Virgil
1/7/13 ross.finlayson@gmail.com
1/7/13 Virgil
1/3/13 fom
1/3/13 fom
1/4/13 mueckenh@rz.fh-augsburg.de
1/4/13 fom
1/5/13 mueckenh@rz.fh-augsburg.de
1/5/13 Virgil
1/5/13 fom
1/6/13 Virgil
1/6/13 fom
1/6/13 Virgil
1/6/13 fom
1/6/13 ross.finlayson@gmail.com
1/4/13 Virgil
1/3/13 mueckenh@rz.fh-augsburg.de
1/3/13 Virgil
1/3/13 forbisgaryg@gmail.com
1/3/13 Virgil
1/4/13 Zaljohar@gmail.com
1/4/13 Virgil
1/4/13 Zaljohar@gmail.com
1/4/13 mueckenh@rz.fh-augsburg.de
1/4/13 fom
1/5/13 mueckenh@rz.fh-augsburg.de
1/5/13 fom
1/5/13 mueckenh@rz.fh-augsburg.de
1/5/13 Virgil
1/5/13 fom
1/5/13 Virgil
1/4/13 Virgil
1/3/13 mueckenh@rz.fh-augsburg.de
1/3/13 Virgil
1/4/13 mueckenh@rz.fh-augsburg.de
1/4/13 fom
1/4/13 Virgil
1/2/13 Bill Taylor